So from the squared slope of the logaritmized tunneling current one can determine the work function average. Since this value is position dependent one can obtain a local-work-functon-image better known as local-barriere-height (LBH) image.

Now here's the problem:

Reversely, varying the current always in the same way, meaning independent of the LBH at the current position, produces different variation amplitudes of delta depending on the local work function. What most current (2018) STMs are using is PI control that uses exactly that position independent constant gain. And this is what regularly gets them into an unstable regime (a regime where the actual current widely detours from desired current) at locations with low LBH. This is leading to the aforementioned "shadows".

Here's what the papers authors did to solve the issue:

They superimposed a "high" frequency dithering signal (dither frequency was 4kHz) onto the unprocessed feedback signal such that they could determine the LBH based on the resulting current variations. (This part was not new.)

Then they use the gained LBH value to continuously (LBH estimation bandwidth was 400Hz)
re-tune the DC gain of the STM's PI controller. Re-tune the the proportional P part. (PI Feedback bandwidth was 300Hz.)

As a side-note: They used some alternative implementation of a lock in amplifier including second order band pass filters and first order Lyapunov filters. They write that they have outlined details about that in one of their preceding papers.

The results:

(Fig. 5.):
(1) Significant reduction of the unwanted correlation of the LBH images with the topography images.

(Fig. 6):
(2) At the usually wanted and or necessary high gain settings near the stability limit, sudden drops in LBH (like in case of dangling bonds) do no longer lead to PI control breakdown. The old "solution" of reducing the overall gain led to more tip-sample crashes (especially in lithography mode) due to less sensitivity and smaller bandwidth.

(Note: The shadows are no crashes. They are more like over-retracts. When lowering DC gain to reduce these shadows, this is when one gets crashes.)

They write that the usual assumption of the "gap modulation method" is that the delta dithering amplitude is constant because the modulating frequency is beyond the controller bandwidth. ("gap modulation method" == established method of feedback dithering for LBH image generation)
They write that this assumption does not always hold. Especially for fast-scanning high-bandwidth scanners.
And I take (my interpretation reading between lines) they mean the problem was not solved till now because this problem was overlooked.

All this was done with big slow macroscopic piezo based STMs.
(An in house own design STM of Zyvex and an omicron STM for comparison.)

So we are left to wonder how much this will do for fast and lightweight MEMS based STMs.